Abstract
We investigate the approximation properties of nonlinear integral operators of the convolution type. In this approximation, we use functions of bounded variation based on the appropriate functionals. To get more general results, we consider Bell-type summability methods in the approximation. Moreover, we examine the rate of approximation. Then, using summability methods, we obtain a characterization for absolute continuity. Our examples at the end of the paper clearly demonstrate why we used summability methods rather than convergence in the conventional sense.
Similar content being viewed by others
References
Angeloni, L.: Approximation results with respect to multidimensional \(\varphi \)-variation for nonlinear integral operators. Z. Anal. Anwend. 32(1), 103–128 (2013)
Angeloni, L., Vinti, G.: Convergence in variation and rate of approximation for nonlinear integral operators of convolution type. Results Math. 49(1–2), 1–23 (2006)
Angeloni, L., Vinti, G.: Approximation by means of nonlinear integral operators in the space of functions with bounded \(\varphi \)-variation. Differ. Integral Equ. 20, 339–360 (2007)
Angeloni, L., Vinti, G.: Convergence and rate of approximation for linear integral operators in \(\text{ BV}^{\varphi }\)-spaces in multidimensional setting. J. Math. Anal. Appl. 349, 317–334 (2009)
Angeloni, L., Vinti, G.: Errata corrige to: approximation by means of nonlinear integral operators in the space of functions with bounded \(\varphi \)-variation. Differ. Integral Equ. 23, 795–799 (2010)
Angeloni, L., Vinti, G.: Convergence and rate of approximation in \(BV_{\varphi }({\mathbb{R}}_{N}^{+}) \) for class of Mellin integral operators. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni 25(3), 217–232 (2014)
Angeloni, L., Vinti, G.: A concept of absolute continuity and its characterization in terms of convergence in variation. Math. Nachr. 289(16), 1986–1994 (2016)
Aslan, İ., Duman, O.: A summability process on Baskakov-type approximation. Period. Math. Hungar. 72(2), 186–199 (2016)
Aslan, I., Duman, O.: Summability on Mellin-type nonlinear integral operators. Integral Transform. Spec. Funct. 30(6), 492–511 (2019)
Aslan, I., Duman, O.: Approximation by nonlinear integral operators via summability process. Math. Nachr. 293(3), 430–448 (2020)
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. Analysis 23, 299–340 (2003)
Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, vol. 9. New York, Berlin (2003)
Bardaro, C., Sciamannini, S., Vinti, G.: Convergence in \(\text{ BV}_{\varphi }\) by nonlinear Mellin-type convolution operators. Funct. Approx. Comment. Math. 29, 17–28 (2001)
Bell, H.T.: \({\cal{A}}\)-summability, Dissertation. Lehigh University, Bethlehem (1971)
Bell, H.T.: Order summability and almost convergence. Proc. Am. Math. Soc. 38, 548–552 (1973)
Bertero, M., Brianzi, P., Pike, E.: Super-resolution in confocal scanning microscopy. Inverse Probl. 3, 195–212 (1987)
Bose, N., Boo, K.: High-resolution image reconstruction with multisensors. Int. J. Imaging Syst. Technol. 9, 294–304 (1998)
Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation. Volume 1: One-Dimensional Theory. Pure Appl. Math. Vol. 40 (Academic Press, New York-London, 1971)
Dawson, D.F.: Matrix summability over certain classes of sequences ordered with respect to rate of convergence, Pacific. J. Math. 24, 51–56 (1968)
Ford, N.J., Edwards, J.T., Frischmuth, K.: Qualitative Behaviour of Approximate Solutions of Some Volterra Integral Equations with Non-Lipschitz Nonlinearity, Numerical Analysis Report 310. Manchester Centre for Computational Mathematics, Manchester (1997)
Gonzalez, R., Woods, R.: Digital Image Processing. Addison-Wesley, Boston (1993)
Hansen, P.C.: Deconvolution and regularization with Topelitz matrices. Numer. Algorithms 29, 323–328 (2002)
Hardy, G.H.: Divergent series. Oxford Univ. Press, London (1949)
Jordan, C.: Sur la serie de Fourier. C. R. Acad. Sci. Paris 92, 228–230 (1881)
Jurkat, W.B., Peyerimhoff, A.: Fourier effectiveness and order summability. J. Approx. Theory 4, 231–244 (1971)
Jurkat, W.B., Peyerimhoff, A.: Inclusion theorems and order summability. J. Approx. Theory 4, 245–262 (1971)
Keagy, T.A., Ford, W.F.: Acceleration by subsequence transformations. Pac. J. Math. 132(2), 357–362 (1988)
Kress, R.: Linear Integral Equations. Springer, Berlin (1989)
Küçük, N., Duman, O.: Summability methods in weighted approximation to derivatives of functions. Serdica Math. J. 41(4), 355–368 (2015)
Laczkovich, M., Sos, V.T.: Functions of bounded variation. In: Real Analysis, Springer, New York, 399-406 (2015)
Lorentz, G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)
Love, E.R., Young, L.C.: Sur une classe de fonctionnelles linéaires. Fund. Math. 28, 243–257 (1937)
Lu, Y., Shen, L., Xu, Y.: Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Probl. 26, 32 (2010)
Maligranda, L., Orlicz, W.: On some properties of functions of generalized variation. Monatsh Math. 104, 53–65 (1987)
Maligranda, L.: On interpolation of nonlinear operators. Comment. Math. Prace. Mat. 28(2), 253–275 (1989)
Matuszewska, W., Orlicz, W.: On property \(\text{ B}_{{1}}\) for functions of bounded \(\varphi \)-variation. Bull. Pol. Acad. Sci. Math. 35(1–2), 57–69 (1987)
Musielak, J., Orlicz, W.: On generalized variations (I). Studia Math. 18, 11–41 (1959)
Mohapatra, R.N.: Quantitative results on almost convergence of a sequence of positive linear operators. J. Approx. Theory 20, 239–250 (1977)
Ng Michael, K.: Circulant preconditioners for convolution-like integral equations with higher-order quadrature rules. Electron. Trans. Numer. Anal. 5, 18–28 (1997)
Radó, T.: Length and Area, American Mathematical Society Colloquium Publications, vol. 30. American Mathematical Society, New York (1948)
Smith, D.A., Ford, W.F.: Acceleration of linear and logarithmical convergence, Siam. J. Numer. Anal. 16, 223–240 (1979)
Swetits, J.J.: On summability and positive linear operators. J. Approx. Theory 25, 186–188 (1979)
Vinti, G.: A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 79(1–2), 217–238 (2001)
Wiener, N.: The quadratic variation of a function and its Fourier coefficients. Mass. J. Math. 3, 72–94 (1924)
Wimp, J.: Sequence transformations and their applications. Academic Press, New York (1981)
Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)
Young, L.C.: Sur une généralisation de la notion de variation de puissance \(\text{ p}^{ieme}\) bornée au sens de M. Wiener, et sur la convergence des séries de Fourier. C. R. Acad. Sci. Paris 204, 470–472 (1937)
Ziemer, W.P.: Weakly differentiable functions: Sobolev spaces and functions of bounded variation. Springer Science & Business Media 120, (2012)
Acknowledgements
The author would like to thank reviewer(s) for insightful comments and reading the manuscript carefully.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Aslan, İ. Convergence in Phi-Variation and Rate of Approximation for Nonlinear Integral Operators Using Summability Process. Mediterr. J. Math. 18, 5 (2021). https://doi.org/10.1007/s00009-020-01623-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01623-2
Keywords
- Summability process
- nonlinear integral operators
- convolution-type integral operators
- bounded \(\varphi \)-variation
- rate of approximation